Correlation Between Muon g-2 and \mu\rightarrow{e}{\gamma}

# Correlation Between Muon g−2 and μ→eγ

Wei-Chi Chiu111ericgreat@gmail.com, Chao-Qiang Geng222geng@phys.nthu.edu.tw, Da Huang333dahuang@phys.nthu.edu.tw Department of Physics, National Tsing Hua University, Hsinchu, Taiwan
Physics Division, National Center for Theoretical Sciences, Hsinchu, Taiwan
Chongqing University of Posts & Telecommunications, Chongqing, 400065, China
July 2, 2019
###### Abstract

While the muon anomaly can be successfully explained by some new physics models, most of them are severely constrained by the bound. This tension is more transparent from the effective field theory perspective, in which the two phenomena are encoded in two very similar operators. However, with the Wilson coefficients, the current upper bound on indicates a new-physics cutoff scale being five orders smaller than that needed to eliminate the anomaly. By summarizing all the formulae from the one-loop contributions to the muon with the internal-particle spin not larger than 1, we point out two general methods to reconcile the conflict between the muon and : the GIM mechanism and the non-universal couplings. For the latter method, we use a simple scalar leptoquark model as an illustration.

## I Introduction

The Standard Model (SM) provides an excellent description of elementary particles and interactions, which has further been confirmed by the discovery of the SM-like Higgs particle at the Large Hadron Collider (LHC) Beringer:1900zz (); Aad:2012tfa (); Chatrchyan:2012ufa (). Nevertheless, there are still some experimental results showing the tantalizing hints to new physics beyond the SM. Especially, one of the biggest discrepancies between the experimental values and the SM predictions comes from the muon anomalous magnetic moment (AMM) or for short, known as the anomaly.

In 2001, the E821 experiment at Brookhaven National Lab (BNL) Bennett:2004pv (); Bennett:2006fi () showed that the measured value of exceeds the SM prediction by about , where the different standard deviations are originated from the various theoretical methods for estimating the hadronic contributions, by applying either the  Davier:2010nc () or the -based Alemany:1997tn () data. The value of the discrepancy between the measurement and SM prediction is given by Beringer:1900zz (); Bennett:2006fi (); Kinoshita:2005sm (); Czarnecki:2002nt (); Hagiwara:2011af (); Davier:2004gb (); Passera:2004bj ()

 Δaμ=aexpμ−aSMμ=287(63)(49)×10−11. (1)

To explain this anomaly, many models have been proposed (see Jegerlehner:2009ry () and references therein). For example, by including the mixings of the muon with some new TeV-scale heavy leptons Czarnecki:2001pv (); Kannike:2011ng (), the new contribution to is roughly the same order as the - and -boson ones due to the enhancement from the heavy lepton masses.

However, most of these models predict visible lepton flavor violating (LFV) processes. In particular, the decay of usually gives one of the most stringent constraints, due to the accuracy of its experimental searches. For example, the MEG collaboration has recently renewed the upper bound on the branching ratio of this process to ( C.L.) Adam:2013mnn (). As a result, it is natural and necessary to use this updated limit to constrain the structure of new physics behind the anomaly.

The main purpose of the present paper is to propose some general methods of building models to not only explain the anomaly but also satisfy the stringent constraint. We first explore this problem from the effective field theory (EFT) perspective, finding that it is very challenging to accomplish both processes simultaneously with the same UV cutoff and the similar order of the Wilson coefficients. We then focus on some simplified models with new physics perturbatively coupled to the SM fields so that the leading order contributions to and appear at the one-loop order. By collecting all one-loop contributions with different spins (), CP properties and electric charges of new particles, we find two general methods to reconcile the tension between the anomaly and the constraint: the GIM mechanism and the non-universal couplings. For the latter, we present a simple UV-complete leptoquark model as an illustration. In our study, we assume that all of the SM contributions to the muon are already well understood and appropriately calculated. The discrepancy mainly arises from new physics. For clarity, we will use to denote the new physics correction to .

This paper is organized as follows. We analyze the correlation between the muon anomaly and from the EFT perspective in Sec. II. In Sec. III, we classify the general new physics models by the leading-order one-loop diagrams for the muon in terms of their different structures and internal particle contents. In Sec. IV, we proposes two general methods: the GIM mechanism and the non-universal couplings, which can resolve the tension between these two processes. Finally, a short summary is given in Sec. V.

## Ii General analysis from effective operators

It is quite natural to consider the constraint on new physics from the LFV mode to explain the anomaly from the EFT perspective since the structure of the leading effective operators for both processes is essentially the same except for that the outgoing muon is replaced by an electron, as illustrated in Fig. 3.

The effective operators for and are given by:

 δLaμeff = eΛ¯μσμν(CL+CR)μFμν, (2) δLμ→eγeff = eΛ′¯eσμν(C′LPL+C′RPR)μFμν, (3)

respectively, where and . At this stage, we differentiate the cutoff scales as and , and the Wilson coefficients of the left and right-handed couplings as and for and , respectively. Note that we have extracted the electromagnetic coupling constant from the Wilson coefficients for the normalization.

We can easily derive the contribution to from Eq. (2), given by

 δaμ=e(CL+CR)Λ/e2mμ. (4)

With this formula, we find that the desired value of can be obtained by taking  GeV with the natural value of .

It is also straightforward to evaluate the branching ratio for the decay process , given by

 B(μ→eγ) =Γ(μ→eγ)Γ(μ→eνμ¯νe) =24π2G2Fm2μ1Λ′2(∣∣C′L∣∣2+∣∣C′R∣∣2), (5)

where denotes the Fermi constant and is used for the normalization. From Eq. (5) and , one obtains

 ≲3.31×10−27 GeV−2. (6)

If we take the Wilson coefficients to be , the cutoff scale should be at least of  GeV.

From the above discussion, we see that there is at least a five-order scale gap between the cutoff needed to solve the anomaly and that required by the constraint. In other words, if we assume that the same new physics contributes to both processes, i.e., as indicated by the anomaly, the predicted branching ratio for the decay is naturally 5 orders larger than the current experimental bound, unless we choose the Wilson coefficients of or even smaller, which are obviously quite unnatural from the general EFT philosophy. Clearly, we come to the conclusion that it is challenging for a natural model to obtain the required contribution to while satisfying the current constraint from the EFT aspect.

## Iii General Results of (g−2)μ from One-Loop Diagrams

Before providing the general methods to reconcile the tension between and the constraint from , let us first see how general new physics models perturbatively coupled to the SM part can solve the anomaly. We shall work in a simplified framework in which only the relevant particles and the renormalizable parts of the Lagrangian related to are given. In this setup, it is enough to only consider the leading one-loop contributions to . For simplicity, we confine the spin of the loop particles to be not larger than 1, but we do not restrict their charges as to keep the discussion general. In this way, the leading one-loop Feynman diagrams can be classified into 4 categories, as explicitly shown in Fig. 2,

according to if the boson running in the loop is a (axial-)vector or (pseudo-)scalar and the photon is emitted from a fermion or boson. We calculate these diagrams and present the final expressions for , which can be compared with those given in the literature Leveille:1977rc (); Jegerlehner:2009ry (); Lynch:2001zs () with different gauges. In particular, we also show the analytic formulae in two useful limits: (i) and (ii) , where () denotes the mass of the additional loop boson (fermion). However, in a complete model, the total contribution to is usually the summation of the two or more diagrams in Fig. 2, so the classification here is only for the convenience of the discussion.

### iii.1 New Vector Boson

Besides the new vector boson , we usually need to introduce some extra fermions with the internal index . The renormalizable lepton-vector-fermion vertex can be written as follows:

 LVint=−¯ℓα{(CV)αiγμ+(CA)αiγμγ5}f′iXμ+h.c., (7)

where the subscript is the charged flavor index and the (axial-)vector coupling matrix. With this setup, Fig. 2 and 2 can be calculated as follows.

#### iii.1.1 Photon Emitted From the Internal Fermions (Fig. 2)

The contributions to due to the vector (V) and axial-vector (A) couplings are given by Jegerlehner:2009ry ()

 δaVμ = −Qf′i|(CV)μi|24π2m2μMV2IV IV = 12∫10dx2x(1−x)(x−2+2ϵ)+λ2(1−ϵ)2x2(1−x+ϵ)(1−x)(1−λ2x)+(ϵλ)2x (8) = −112(1−z)4(8−38z+39z2−14z3+5z4−18z2lnz) +ϵ4(1−z)3(4−3z−z2+6zlnz),ifMV∼Mf′i≫mμ

and

 δaAμ = −Qf′i|(CA)μi|24π2m2μMV2IA IA = 12∫10dx2x(1−x)(x−2−2ϵ)+λ2(1+ϵ)2x2(1−x−ϵ)(1−x)(1−λ2x)+(ϵλ)2x (9) = −112(1−z)4(8−38z+39z2−14z3+5z4−18z2lnz) −ϵ4(1−z)3(4−3z−z2+6zlnz),ifMV∼Mf′i≫mμ

respectively, where is the mass of the vector boson, , , and is the electric charge of . If we further restrict to the limit of , the two integrals, and , can be simplified to

 IV = ϵ−23, IA = −ϵ−23. (10)

As a simple check of our general formulae, Eqs. (III.1.1) and (III.1.1), we can identify as the boson and as the lepton, resulting in

 δaZμ=δaVμ+δaAμ (11)

which agrees with the usual SM calculations in the literature Beringer:1900zz (). Here, we have used , , , , and .

Furthermore, since the integral for the purely axial-vector coupling is always negative, the axial-vector contribution from negative charged fermions in the loop is always desconstructive with the SM one. Clearly, there is no hope to add a neutral vector boson with the purely axial-vector coupling to explain the anomaly.

#### iii.1.2 Photon Emitted From A Charged Vector Boson (Fig. 2)

In order to differentiate from the previous case, we denote the two general contributions to as those from the charged vector (CV) and charged axial-vector (CA) couplings, given by Jegerlehner:2009ry ()

 δaCVμ = −QV|(CV)μi|24π2m2μMV2ICV ICV = 12∫10dx2x2(1+x−2ϵ)−λ2(1−ϵ)2x(1−x)(x+ϵ)x+(ϵλ)2(1−x)(1−ϵ−2x) (12) = 112(1−z)4(10−43z+78z2−49z3+4z4+18z3lnz) −ϵ4(1−z)3(4−15z+12z2−z3−6z2lnz),ifMV∼Mf′i≫mμ,

and

 δaCAμ = −QV|(CA)μi|24π2m2μMV2ICA ICA = 12∫10dx2x2(1+x+2ϵ)−λ2(1+ϵ)2x(1−x)(x−ϵ)x+(ϵλ)2(1−x)(1−ϵ−2x) (13) = 112(1−z)4(10−43z+78z2−49z3+4z4+18z3lnz) +ϵ4(1−z)3(4−15z+12z2−z3−6z2lnz),ifMV∼Mf′i≫mμ,

respectively, where is the vector boson electric charge. In the limit of , e.g., and have the following simple form

 ICV = −ϵ+56, ICA = ϵ+56. (14)

We can check Eqs. (III.1.2) and (III.1.2) easily by identifying the charged boson to be the W boson with and , giving

 δaWμ=δaCVμ+δaCAμ=GFm2μ8√2π2103, (15)

which is in agreement with the usual SM result at one-loop level Beringer:1900zz ().

Note that there are no cross terms of the vector and axial-vector couplings, which are proportional to or in the contribution, since such terms lead to the muon electric dipole operator, rather than the magnetic dipole one which we are interested in. In addition, our derivation is based on the Lagrangian of Eq. (7), in which the vector boson coupling is decomposed into the vector and axial-vector couplings. However, it is more useful to work on the basis in which the fermion field has definite chirality. The problem is how to use our general formulae Eqs. (III.1.1), (III.1.1), (III.1.2) and (III.1.2) in this chiral basis. We find that if the two vertices involving are both left-handed or both right-handed, i.e., the internal fermion does not flip its chirality, the expression should be . But, if one vertex is left-handed while the other is right-handed, the internal fermion flips its chirality with the result of .

### iii.2 New Scalar/Pseudoscalar Boson

For new scalar (S) and pseudoscalar (P) bosons, the couplings to the SM leptons require to have some additional fermions through the following Yukawa couplings:

 LSint = Yαi¯ℓαSf′i+h.c., (16) LPint = Y′αi¯ℓαiγ5Pf′i+h.c., (17)

where and are Yukawa coupling matrices for scalars and pseudoscalars, respectively, and denotes the lepton flavor. The relevant Feynman diagrams are depicted in Figs. 2 and 2.

#### iii.2.1 Photon emitted from internal fermion (Fig. 2)

In order for this diagram to contribute to , the internal fermions should have charge , no matter if the (pseudo)scalar is charged or not. Moreover, it is convenient to further divide the contribution into two parts: scalar (S) and pseudoscalar (P) ones, since they lead to the different expressions.

Scalar: The general contribution to from a scalar (S) coupling in Eq. (16) is

 δaSμ = −Qf′i|Yμi|24π2m2μMS2IS IS = 12∫10dxx2(1−x+ϵ)(1−x)(1−λ2x)+(ϵλ)2x (18) = 112(1−z)4(2+3z−6z2+z3+6zlnz) −ϵ4(1−z)3(3−4z+z2+2zlnz),ifMS∼Mf′i≫mμ,

where is the scalar mass, , , , and is the charge of as before. In the limit of , we can further simplify the integral to

 IS = −ϵln⎛⎝Mf′iMS⎞⎠−34ϵ+16. (19)

Note that the integral in Eq. (III.2.1) does not have the definite sign in the parameter region we are interested in. As a result, it is more useful to plot against and on the 3D diagram in Fig. 3(a). We find that in the limits and , the value of is always positive. Thus, the sign of only depends on . For example, if we take the fermion charge to be and the scalar to be neutral, the new contribution to the muon is constructive with the SM one. We can obtain by choosing and  TeV, promising to be probed by the next run of the LHC experiments.

Pseudoscalar: The pseudoscalar (P) coupling in Eq. (17) can give the following general contribution,

 δaPμ = −Qf′i|Yμi|24π2m2μMP2IP IP = 12∫10dxx2(1−x−ϵ)(1−x)(1−λ2x)+(ϵλ)2x (20) = 112(1−z)4(2+3z−6z2+z3+6zlnz) +ϵ4(1−z)3(3−4z+z2+2zlnz),ifMP∼Mf′i≫mμ,

where denotes the pseudoscalar mass, , , and . If we further restrict to the limit of , the integral can be reduced to

 IP = (21)

As for the scalar case, we also plot in Eq. (III.2.1) against and on the 3D diagram in Fig. 3(b). It is useful to note that most of the parameter space is negative except for the region.

As an example, if and the pseudoscalar is neutral, the sign of its contribution to the muon is positive only when , i.e., , as seen from Fig. 3(b). However, if there exists such a light charged fermion, it should have already been observed at the colliders, such as the LEP. Therefore, a model with a neutral pseudoscalar and a light charged fermion favored by the current data is already ruled out.

#### iii.2.2 Photon emitted from charged scalar or pseudoscalar particles (Fig. 2)

The calculation of the Feynman diagram in Fig. 2 leads to the contribution to the muon AMM from loops with a charged scalar (CS) or pseudoscalar (CP).

Scalar: When the photon is emitted from a charged scalar, the general contribution to the muon in Fig. 2 is

 δaCSμ = −QS|Yμi|24π2m2μMS2ICS ICS = 12∫10dxx(x−1)(x+ϵ)x+(ϵλ)2(1−x)(1−ϵ−2x) (23) = −112(1−z)4(1−6z+3z2+2z3−6z2lnz) −ϵ4(1−z)3(1−z2+2zlnz),ifMS∼Mf′i≫mμ,

where is the scalar charge, , and as before. When , can be simplified to

 ICS = −14ϵ−112. (24)

The integral in Eq. (III.2.2) is always negative. Consequently, if is minus, will depress . The simple application of this result is that when the loop in Fig. 2 is enclosed by a scalar with and a neutral fermion, the contribution can never reduce the tension between the SM prediction and data for the muon .

Pseudoscalar: When the scalar in Fig. 2 is replaced by the charged pseudoscalar, the expression for the contribution becomes,

 δaCPμ = −QP|Y′μi|24π2m2μMP2ICP ICP = 12∫10dxx(x−1)(x−ϵ)x+(ϵλ)2(1−x)(1−ϵ−2x) (26) = −112(1−z)4(1−6z+3z2+2z3−6z2lnz) +ϵ4(1−z)3(1−z2+2zlnz),ifMP∼Mf′i≫mμ,

where is the charge of the pseudoscalar, , and . The integral in the limit becomes

 ICP = 14ϵ−112. (27)

For a rough estimation, with a pseudoscalar with and a neutral fermion in the loop, the Feynman diagram in Fig. 2 can solve the muon anomaly with and  TeV.

Similar to the (axial-)vector case in the previous subsection, the cross terms of the scalar and pseudoscalar couplings, which are proportional to or , do not contribute to the muon . Rather, they give rise to the muon electric dipole moment. Moreover, when the scalar/pseudoscalar couplings in Eqs. (16) and (17) are decomposed into the basis in which the chiralities of the fermion fields are well defined, we can easily obtain the formulae from Eqs. (III.2.1), (III.2.1), (III.2.2) and (III.2.2). Concretely, when the two vertices on the fermion line are both left- and right-handed, the result would be . Note that the internal fermion has to flip its chirality in this case so that the final expression should be proportional to the internal fermion mass. However, for the case that the two couplings have different chiralities, the result is .

It is interesting to note that our general formalism can be applied to the general supersymmetric extension of the SM. One particular example is the minimal supersymmetric Standard Model (MSSM), in which neutralinos, charginos and various sleptons provide the additional contributions to and . Especially, one-loop diagrams related to charginos and neutralinos correspond to our case in Sec. III.2.1 and III.2.2, so the formulae in these two subsections can be used directly. The detailed discussions of and its correlation to are summarized in Ref. Stockinger:2006zn ().

## Iv Some General Solutions to the Tension Between the (g−2)μ anomaly and the μ→eγ Constraint

Now, we come back to the question how to reconcile the tension between the muon anomaly and the constraint in a generic new physics model. Here, we provide two generic methods mentioned in Sec. I.

### iv.1 GIM Mechanism

As is well known in the SM, there is no flavor-changing neutral current (FCNC) at tree level due to the famous GIM mechanism Glashow:1970gm (). Even at loop level, the GIM mechanism also suppresses the FCNC greatly. Of our present interest is the SM contributions to and . In the SM with the non-zero neutrino masses, the one for can be obtained by the loops running a -boson with different flavors of neutrinos. But because of the GIM mechanism, the unitarity of the PMNS matrix results in the leading terms to be canceled. The terms left are at least proportional to the square of the neutrino masses. Since the neutrino masses are negligibly small, it is no hope to observe the rate in the current experiments. However, does not suffer such a huge suppression, leaving us measurable signals. We hope that the GIM mechanism may also happen in the new physics sector beyond the SM.

To be specific, let us consider a model in which a vector and some fermions with masses are introduced with the following chiral couplings:

 L=g¯ℓαUαiγμPLfiXμ+h.c., (28)

where the different charged leptons in the SM with the flavor index . We have extracted the overall coupling constant to make the mixing matrix to satisfy the following orthogonal normalization conditions,

 ∑iUαiU∗βi ={1,if α=β.0,if α≠β., (29)

where and are the flavor indices of charged leptons (, , ). Here, the mixing matrix is not necessarily unitary but can be extended to a rank-three one, where () is the number of the internal fermions. Note that the Lagrangian in Eq. (28) is very similar to the SM W-boson couplings with the W-boson replaced by and the neutrinos by . However, to make the discussion more general, we take the charge of the vector boson to be so that the fermion charge should be . We also assume a mass hierarchy () for simplicity.

With the Lagrangian in Eq. (28), the leading-order one-loop Feynman diagrams relevant to are essentially the same as the first two diagrams in Fig. 2 except for the outgoing replaced by , resulting in the following effective operator,

 δLμ→eγeff=¯eσμνPRμFμν(g2mμ)∑iU∗μiUei¯C(mX,mi,me,mμ) (30)

where

 ¯C(mX,mi,me,mμ)={QXI1(mX,mi,me,mμ)+QiI2(mX,mi,me,mμ)}, (31)

and the integrals are the separated expressions for the two Feynman diagrams in Fig. 2. The explicit expressions of are not given since they are not very crucial for our discussion of the general GIM mechanism. What we should focus is the track of the coupling factors, especially the mixing matrix elements , which can be easily read out from the Feynman diagrams. Note that the two external charged leptons in the effective vertex have different chiralities, while the interaction in Eq. (28) only involves left-handed fermions. Therefore, an extra lepton mass insertion is needed to obtain by flipping the chirality of either lepton, producing two terms proportional to and . However, since , only the latter term is kept, which is the origin of the factor and the right-handed projection operator in Eq. (30). Moreover, the mass dimensions of the integrals are , which can be made dimensionless by extracting the inverse of the largest mass scale squared. In this way, the total integral can be written in the following form:

 ¯C(mX,mi,me,mμ) = 1m2XC(mi/mX,me/mX,mμ/mX) (32) = 1m2X(C0+C2m2im2X+...),

where are the coefficients for the power expansion of in terms of , and are usually expected as the (1) function of . Since only the squares of various particle masses appear in the integral , the expansion only has even powers of .

By putting Eq. (32) back into Eq. (30), the effective operator for has the form:

 δLμ→eγeff=¯eσμνPRμFμν(g2mμm2X)∑iU∗μiUei(C0+C2m2im2X+...). (33)

Note that the leading term vanishes due to the orthogonal condition in Eq. (29). We are, then, left with the second-order term

 δLμ→eγeff=¯eσμνPRμFμν(g2mμm2X)∑iU∗μiUei{C2m2im2X+O(m4im4X)}, (34)

since, in general, we have

 ∑iU∗μiUei m2i≠0, (35)

for the different internal fermion masses. By considering the hierarchy , we see a large suppression at least in order of . In contrast, for the flavor conserving muon correction, there is no such suppression due to the normalization condition for the outgoing and incoming leptons with the same flavor. Typically, we expect that the contribution to should be of order:

 δLaμeff=¯μσμνμFμν(g2mμm2X)(C′0+O(m2im2X)), (36)

with some (1) coefficient . Thus, the LFV process of is naturally suppressed greatly compared with .

Note that the GIM mechanism is also applicable to the case when the vector interaction is purely right-handed, rather than left-handed as discussed above. However, if the couplings of the new vector to the leptons and the internal fermions have both left-handed and right-handed parts, the GIM mechanism generically breaks down. One reason is that the couplings of different chiralities involve different mixing matrices. For example, let us denote the left-handed vertex mixing matrix by , while the right-handed by . Obviously, the products of two matrices, such as and , do not necessarily obey the normalized orthogonal conditions in Eq. (29). Furthermore, when the two vertices in the diagrams shown in Figs. 2(a) and 2(b) have different chiralities, there should be an additional internal fermion mass insertion for flipping its chirality, rather than the much smaller lepton masses. As a result, the leading-order new-physics corrections to and should be of the same magnitude. Consequently, we can encode this argument in terms of the following effective operators

 δLaμeff = (gLgRm2X)∑i¯μσμν(U∗μiVμiPL+V∗μiUμiPR)μFμνmi{C′0+O(m2im2X)}, δLμ→eγeff = (gLgRm2X)∑i¯eσμν(U∗μiVeiPL+V∗μiUeiPR)μFμνmi{C0+O(m2im2X)